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1.1Of or denoting the system of geometry based on the work of Euclid and corresponding to the geometry of ordinary experience.
‘It reduced the problem of consistency of the axioms of non-Euclidean geometry to that of the consistency of the axioms of Euclidean geometry.’
‘Today we call these three geometries Euclidean, hyperbolic, and absolute.’
‘He was one of the earliest mathematicians to demonstrate that the ordinary experience of Euclidean concepts can be extended meaningfully beyond geometry into the idealised constructions of more complex abstract mathematics.’
‘The falseness of the idea of principle, is typified by a Cartesian or Euclidean geometry.’
‘From this point of view, Euclidean geometry is a very favorable place to begin a student's serious mathematical training.’
‘Well, look at Cartesian geometry: In a Cartesian geometry - or Euclidean, which are interchangeable, in one sense - you have certain assumptions.’
‘The first two volumes cover the foundations of Euclidean geometry and the introduction of a coordinate system, volume 3 studies solid geometry considering quadrics, cubic curves in space, and cubic surfaces.’
‘The approach that concentrates on non-Euclidean geometry is ideal for students who already have a mastery of Euclidean geometry, but it cannot replace such a mastery.’
‘For example, in Euclidean geometry, the relevant invariants are embodied in quantities that are not altered by geometric transformations such as rotations, dilations, and reflections.’
‘This is, of course, how Beltrami first showed that hyperbolic geometry was no less consistent than Euclidean geometry (though he used a different model).’
‘To sum up, I am asserting that Euclidean geometry is the only mathematical subject that is really in a position to provide the grounds for its own axiomatic procedures.’
‘In 1869, after Beltrami's letter… he realized he had made a mistake: the empirical concept of a rigid body and mathematics alone were not enough to characterize Euclidean geometry.’
‘Recently I have decided to capitulate and adopt Isaacs, which shuns both axiomatics and hyperbolic geometry in favor of actual problem solving and construction problems in standard Euclidean geometry.’
‘Similarly, an eliminative structuralist account of real analysis and Euclidean geometry requires a background ontology whose cardinality is at least that of the continuum.’
‘A Euclidean geometry is based on false assumptions, which are called definitions, axioms, and postulates.’
‘The Vertical Angles Conflict Activity was designed for students about to embark upon the study of Euclidean geometry with reference to formal definitions and proofs in class.’
‘If all accelerated systems are equivalent, then Euclidean geometry cannot hold in all of them.’
‘This is a peer reviewed journal devoted to the Euclidean geometry.’
‘The second chapter presents a development of absolute and Euclidean geometry based on Hilbert's axioms.’
‘Conservative mathematicians maintained that such concepts would call into question the very existence and permanence of mathematical truth, as so nobly represented by Euclidean geometry.’
1.2Of such a nature that the postulates of Euclidean geometry are valid.
‘But that law assumes the conservation of mass energy as well as a space which is Euclidean.’
‘This indicates that a simple Euclidean distance in feature space can be used to quantify the relative similarity between different mutant types.’
‘Mathematics has considered alternatives to Euclidean space since the early nineteenth century.’
‘The latter are mediated by DNA-loops bringing two chemically remote segments of the DNA close in Euclidean space.’
‘Humans made a mark without destroying nature, enhancing it by making a Euclidean statement on the raw wilderness, which made its mysteries more awesome and gave it dimension, direction, making it comprehensible.’
‘Mapping it onto the Earth's surface is far more complex, however, because there may be little relationship between proximity in Euclidean geographic space and positionality.’
‘Not that the propositions of geometry are only approximately true, but that they remain absolutely true in regard to that Euclidean space which has been so long regarded as being the physical space of our experience.’
‘By that I mean that they would be chunks of familiar Euclidean space; one could require them to be cuboids, but this is not very important mathematically.’
‘These large matrices describe high-dimensional Euclidean spaces within which biomolecular sequences can be uniquely represented as vectors.’
‘It is important to note, however, that the extra dimension, though curled up, was still Euclidean in nature.’
‘But why does space have to be Euclidean, nice and flat and square.’
‘Now within any bounded region of Euclidean space it can be shown that Cantor's continua coincide with continua in the sense of the modern definition.’
‘If physical space and perceptual space are the same thing, then Kant is claiming we know a priori that physical space is Euclidean.’
‘Then like in the linear separable case, it finds the optimal separating hyper-plane in the Hilbert space H that would correspond to a nonlinear boundary in the original Euclidean space.’
‘In this case, one would like good algorithms for embedding specifically into 2-and 3-dimensional Euclidean spaces.’
‘Obviously, when no obstacles are used, then the matrix represents a Euclidean space with dimensionality equal to two.’
‘However, classical space is Euclidean, and by definition.’
‘Multidimensional scaling analyses were used to represent the relationships of the data set in n-dimensional Euclidean space in an attempt to identify putative group structures.’
‘It seemed to me that I could do some useful work in giving the student a historical perspective and in showing how the multitude of abstract concepts have arisen and are present in Euclidean spaces.’
‘A hyperbolic display contains much more space than a simple Euclidean plane because the circumference and area of the circle it's mapped upon grows exponentially with the length of its radius.’